Subgroup and quotient

Algebra Level 4

Let GG be a group, and NN a normal subgroup. Which of the following statements is/are always true?

I. If NN is finite and G/NG/N is finite, then GG is finite.
II. If NN is finite and cyclic and G/NG/N is finite and cyclic, then GG is finite and cyclic.
III. If NN is abelian and G/NG/N is abelian, then GG is abelian.


Notation:

  • A finite cyclic group is a group that is isomorphic to Zn, {\mathbb Z}_n, the integers mod n,n, for some n.n.
  • An abelian group is a group whose operation is commutative: xy=yxx * y = y * x for all x,yG.x,y \in G.
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